Exact Liouvillian Spectrum of a One-Dimensional Dissipative Hubbard Model

Masaya Nakagawa,1, ∗ Norio Kawakami,2 and Masahito Ueda1, 3, 4

1Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan2Department of Physics, Kyoto University, Kyoto 606-8502, Japan

3RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan4Institute for Physics of Intelligence, University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Dated: March 26, 2021)

A one-dimensional dissipative Hubbard model with two-body loss is shown to be exactly solvable.We obtain an exact eigenspectrum of a Liouvillian superoperator by employing a non-Hermitianextension of the Bethe-ansatz method. We find steady states, the Liouvillian gap, and an exceptionalpoint that is accompanied by the divergence of the correlation length. A dissipative version of spin-charge separation induced by the quantum Zeno effect is also demonstrated. Our result presents anew class of exactly solvable Liouvillians of open quantum many-body systems, which can be testedwith ultracold atoms subject to inelastic collisions.

In quantum physics, no realistic system can avoid thecoupling to an environment. The problem of decoherenceand dissipation due to an environment is crucial even forsmall quantum systems. Furthermore, recent remarkableprogress in quantum simulations with a large numberof atoms, molecules, and ions has raised a fundamentaland practical problem of understanding open quantummany-body systems, where interparticle correlations areessential [1–4]. Within the Markovian approximation,the nonunitary dynamics of an open quantum system isgenerated by a Liouvillian superoperator acting on thedensity matrix of the system [5–7]. While interestingsolvable examples have been found [8–18], the diagonal-ization of a Liouvillian of a quantum many-body systemis more challenging than that of a Hamiltonian. Extend-ing the class of exactly solvable models to the realm ofdissipative systems and discovering prototypical solvablemodels that can be realized experimentally should pro-mote the deepening of our understanding of strongly cor-related open quantum systems.

The Hubbard Hamiltonian provides a quintessentialmodel in quantum many-body physics, where the inter-play between quantum-mechanical hopping and interac-tions plays a key role. In particular, equilibrium prop-erties of the one-dimensional case are well understoodwith the help of the exact solutions [19–21]. The Hub-bard model has been experimentally realized with ultra-cold fermionic atoms in optical lattices [22], and the highcontrollability in such systems has recently invigoratedthe investigation of the effect of dissipation due to par-ticle losses [23]. In this Letter, we show that the one-dimensional Hubbard model subject to two-body parti-cle losses is exactly solvable. On the basis of the ex-act solution, we obtain an eigenspectrum of the Liou-villian, and elucidate how dissipation fundamentally al-ters the physics of the Hubbard model. Our main find-ings are threefold. First, we obtain the exact steadystates and long-lived eigenmodes that govern the relax-ation dynamics after a sufficiently long time. Second, we

show that excitations above the Hubbard gap are sig-nificantly altered by dissipation and find that the modelshows novel critical behavior near an exceptional point[24] that originates from the nondiagonalizability of theLiouvillian. Third, we demonstrate that spin-charge sep-aration, which is a salient feature of one-dimensional sys-tems [25], is extended to dissipative systems by exploitingthe fact that the strong correlation is induced by dissi-pation even in the absence of an interaction. Our resultshows that a number of exactly solvable Liouvillians canbe constructed from quantum integrable models subjectto loss.Setup.– We consider an open quantum many-body sys-

tem described by a quantum master equation in theGorini-Kossakowski-Sudarshan-Lindblad form [5–7]

dρ

dτ= −i[H, ρ] +

L∑j=1

(LjρL†j −

1

2{L†jLj , ρ}) ≡ Lρ, (1)

where ρ(τ) is the density matrix of a system at timeτ . The system Hamiltonian H is given by the Hubbardmodel on an L-site chain

H = −tL∑j=1

∑σ=↑,↓

(c†j,σcj+1,σ + H.c.) +UL∑j=1

nj,↑nj,↓, (2)

where cj,σ is the annihilation operator of a spin-σ fermion

at site j, and nj,σ ≡ c†j,σcj,σ. The Lindblad operatorLj =

√2γcj,↓cj,↑ describes a two-body loss at site j with

rate γ > 0, which is caused by on-site inelastic collisionsbetween fermions as observed in cold-atom experiments[23, 26–28]. The formal solution of the quantum masterequation can be written down in terms of the eigensystemof the Liouvillian superoperator L defined in Eq. (1). Inthis Letter, we aim at diagonalizing the Liouvillian andobtain exact results for the effect of dissipation on corre-lated many-body systems.Diagonalization of the Liouvillian.– The one-

dimensional Hubbard model, Eq. (2), is known to

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1420

2v2

[co

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be solvable with the Bethe ansatz [19–21]. Here, wegeneralize the solvability of the Hubbard Hamiltonian tothat of the Liouvillian on the basis of the existence ofa conserved quantity in the Hamiltonian [29]. We firstdecompose the Liouvillian into two parts as L = K + J ,where Kρ ≡ −i(Heffρ − ρH†eff) and J ρ ≡

∑Lj=1 LjρL

†j .

The effective non-Hermitian Hamiltonian Heff is givenby Heff ≡ H − i2

∑Lj=1 L

†jLj , and its explicit form is

obtained by replacing U in H with U − iγ, therebymaking the interaction strength complex-valued [30–36].Notably, the one-dimensional Hubbard model with acomplex-valued interaction strength is still integrable[12, 18, 33]. If the interaction strength becomes complex-valued, the SO(4) symmetry of the Hubbard Hamiltonian[37–39] remains intact. In particular, an eigenstate ofthe non-Hermitian Hubbard model can be labeled by thenumber of particles. Let |N, a〉R be a right eigenstateof Heff with N particles: Heff |N, a〉R = EN,a |N, a〉R,where a distinguishes the eigenstates having the sameparticle number. Then, one can diagonalize the super-

operator K as K%(N,n)ab = λ(N,n)ab %

(N,n)ab , where λ

(N,n)ab ≡

−i(EN,a − E∗N+n,b) and %(N,n)ab ≡ |N, a〉R R 〈N + n, b|.

The superoperator J lowers the particle number butnever increases it. Thus, in the representation with

the basis {%(N,n)ab }N,a,b, the Liouvillian L is a triangularmatrix that can easily be diagonalized. This is ageneral property of Liouvillians of systems with loss [29].Indeed, because the eigenvalues of a triangular matrixare given by its diagonal elements, the eigenvalues of the

Liouvillian are given by λ(N,n)ab . The corresponding right

eigenoperator is given by a linear combination of the

basis as C(N,n)ab %

(N,n)ab +

∑N−2N ′=0

∑a′,b′ D

(N,N ′,n)aba′b′ %

(N ′,n)a′b′ ,

where the coefficients D(N,N ′,n)aba′b′ are obtained from

the matrix elements L 〈N ′ − 2, r|Lj |N ′, r′〉R of theLindblad operator Lj with |N ′, r〉L being the lefteigenstate dual to |N ′, r〉R [29, 40]. We thus concludethat if the non-Hermitian Hubbard Hamiltonian Heffis integrable, the Liouvillian L is solvable. Note thatthis does not mean that the Liouvillian itself has anintegrable structure such as the Yang-Baxter relation.Therefore, the mechanism of the solvability here isdifferent from those of previous works on Yang-Baxterintegrable Liouvillians [12, 16–18].

Steady states.– A steady state of the system is char-acterized by an eigenoperator of L with zero eigenvalue.If a state |Ψ〉 is a right eigenstate of Heff with a realeigenvalue, one can show Lj |Ψ〉 = 0, and hence |Ψ〉 〈Ψ|is a steady state [40]. For example, the fermion vacuum|0〉 〈0| is trivially a steady state. Also, in the Hilbert sub-space with no spin-down particles, all eigenstates of Heffcoincide with those in the noninteracting (U = γ = 0)case and thus describe steady states. By letting the spinlowering operator act on the spin-polarized eigenstates,one can construct many steady states owing to the spinSU(2) symmetry of Heff , reflecting the fact that magneti-

zation is conserved during the dynamics [41, 42]. Clearly,these steady states are ferromagnetic and far from thethermal equilibrium states of the one-dimensional Hub-bard model. Physically, the steadiness of the ferromag-netic states can be understood from the Fermi statis-tics, because the spin wavefunction that is fully sym-metric with respect to a particle exchange requires an-tisymmetry in the real-space wavefunction and forbidsdoubly occupied sites that cause a decay, as observed inRefs. [35, 43]. In general, a steady state realized after atime evolution becomes a statistical mixture of the abovesteady states that depends on the initial condition.Bethe ansatz.– We use the Bethe ansatz to obtain

the eigenspectrum of the non-Hermitian Hubbard modelHeff . The Bethe equations are [19–21]

kjL = Φ + 2πIj −M∑β=1

Θ

(sin kj − λβ

u

), (3)

−N∑j=1

Θ

(sin kj − λα

u

)= 2πJα +

M∑β=1

Θ

(λα − λβ

2u

),

(4)

where N is the number of particles, M is the numberof down spins, kj (j = 1, · · · , N) is a quasimomentum,λα (α = 1, · · · ,M) is a spin rapidity, u ≡ (U − iγ)/(4t)is a dimensionless complex interaction coefficient, andΘ(z) ≡ 2 arctan z. The quantum number Ij takes an in-teger (half-integer) value for even (odd) M , and Jα takesan integer (half-integer) value for odd (even) N − M .Here we employ a twisted boundary condition cL+1,σ =e−iΦc1,σ for later convenience, but basically set Φ = 0(i.e., the periodic boundary condition) unless otherwisespecified.Liouvillian gap.– The late-stage dynamics of the sys-

tem near a steady state is governed by long-lived eigen-modes whose eigenvalues are close to zero [44]. By con-struction of the steady states, the long-lived eigenmodescorrespond to Bethe eigenstates in the M = 1 case andtheir descendants derived from the spin SU(2) symmetry.They consist of ferromagnetic spin-wave-type excitations,and their dispersion relation is obtained by a standardcalculation with the Bethe ansatz [40]. Taking consecu-tive charge quantum numbers Ij = −(N+1)/2+j, whichexpress the simplest situation of charge excitations fromthe Fermi surface, we obtain an analytic expression forthe dispersion relation of the spin excitations,

∆E ' − tπu

(Q0 −

1

2sin 2Q0

)(1− cos π∆P

Q0

)(5)

for the momentum ∆P ' 0, where Q0 = πN/L is theFermi momentum. Since the momentum is discretized inunits of 2π/L, the gapless quadratic dispersion around∆P = 0 leads to the smallest imaginary part of the exci-tation energy |Im[∆E]| proportional to 1/L2. Thus, the

3

Liouvillian gap, which is defined by the largest nonzeroreal part of eigenvalues of the Liouvillian, vanishes in thethermodynamic limit, implying a power-law time depen-dence in the decay dynamics [44].

Hubbard gap, correlation length, and exceptionalpoint.– Next, we consider the half-filling case (L = N =2M) and focus on the solution that can be adiabaticallyconnected to the ground state of the Hermitian Hubbardmodel in the limit of γ → 0. Such a solution may not con-tribute to the late-stage behavior due to a short lifetime,but it can be used to study the early-time decay dynam-ics of a Mott insulator. We here assume that U > 0and N (M) is even (odd), and set Ij = −(N + 1)/2 + jand Jα = −(M + 1)/2 + α as in the Hermitian case. Inthe thermodynamic limit, the Bethe equations, Eqs. (3)and (4), reduce to the integral equations for distributionfunctions ρ(k) and σ(λ) as

ρ(k) =1

2π+ cos k

∫Sdλa1(sin k − λ)σ(λ), (6)

σ(λ) =

∫Cdka1(sin k − λ)ρ(k)−

∫Sdλ′a2(λ− λ′)σ(λ′),

(7)

where an(z) ≡ (1/π)[nu/(z2 + n2u2)], and C and S de-note the trajectories of quasimomenta and spin rapidities,respectively [21]. Figure 1 (a),(b) show typical distribu-tions of {kj}j=1,··· ,N and {λα}α=1,··· ,M that are obtainedfrom the solution of the Bethe equations, Eqs. (3) and (4).The distributions indicate that if the trajectories C and Sdo not enclose a pole in the integrands of Eqs. (6) and (7),the trajectories can continuously be deformed to those ofthe γ = 0 case, i.e., C = [−π, π] and S = (−∞,∞). Thus,we obtain the eigenvalue E0 in the thermodynamic limitfrom analytic continuation of the solution in the γ = 0case [19] as

E0/L = −2t∫ ∞−∞

dωJ0(ω)J1(ω)

ω(1 + e2u|ω|), (8)

where Jn(x) is the nth Bessel function. Similarly, theHubbard gap ∆c [19, 45] is given as

∆c = 4tu− 4t[1−

∫ ∞−∞

dωJ1(ω)

ω(1 + e2u|ω|)

]. (9)

Here E0 and ∆c take complex values in general. The life-time of an eigenmode can be extracted from the imag-inary part of the eigenvalue. The absolute value ofIm[E0] ≤ 0 first increases with increasing γ, takes themaximum at some point, and then decreases [40]. Thedecreasing behavior at large γ is attributed to the contin-uous quantum Zeno effect [26, 27, 46–49], which preventsthe creation of doubly occupied sites in eigenstates due toa large cost of the imaginary part of energy. By contrast,the absolute value of Im[∆c] ≤ 0 monotonically increaseswith increasing γ [40], since the excitation corresponding

+

+

- 3 - 2 - 1 1 2 3Re[kj]

-1.0

-0.5

0.5

1.0Im[kj]

+

+

- 3 - 2 - 1 1 2 3Re[λα]

-2-1

12Im[λα]

+

+

- 3 - 2 - 1 1 2 3Re[kj]

-0.6-0.3

0.30.6Im[kj]

+

+

- 2 - 1 1 2Re[λα]

-1.2-0.6

0.61.2Im[λα]

(a) (b)

(c) (d)

FIG. 1. Numerical solutions of the Bethe equations, Eqs. (3)and (4), for L = N = 2M = 250. (a),(c) Blue dots showquasimomenta {kj}, and red crosses show the locations ofpoles at k = ±π − arcsin(±iu). (b),(d) Green dots show spinrapidities {λα}, and red crosses show the locations of polesat λ = ±2iu. The interaction strength is set to u = 1 − 0.5i[(a),(b)] and u = 0.6−0.469i [(c),(d)]. Points on the real axisshow the solutions for the case of γ = 0 at the same U forcomparison.

to the Hubbard gap creates doubly occupied sites. Asthe Liouvillian eigenvalues appear as poles of a single-particle Green’s function [40, 50], the dependence of theHubbard gap on dissipation can be found from the lin-ear response of the dynamics by, e.g., lattice modulationspectroscopy [40, 51, 52].

To further elucidate the physics of the dissipative Mottinsulator, we calculate the correlation length ξ of theabove eigenstate from the asymptotic behavior of the

charge stiffness as∣∣∣d2E0(Φ)dΦ2 |Φ=0∣∣∣ ∼ exp[−L/ξ] (L → ∞)

[53]. The correlation length quantifies the dependence ofthe dynamics on the boundary condition and thus mea-sures the spatial correlation in the eigenmode. We findthat the correlation length is obtained from the analyticcontinuation of the result for the γ = 0 case [53]:

1

ξ= Re

[1

u

∫ ∞1

dyln(y +

√y2 − 1)

cosh(πy/2u)

]. (10)

Figure 2 (a)-(c) show the correlation length for differentvalues of the repulsive interaction. For large γ, the corre-lation length decreases in all cases, indicating that parti-cles are more localized due to dissipation. This behavioris consistent with the quantum Zeno effect [26, 27, 46–49]. On the other hand, when U is small, the correla-tion length grows at an intermediate dissipation strength[see Fig. 2 (b)], implying that dissipation facilitates thedelocalization of particles. Surprisingly, the correlationlength even diverges for small U and takes negative valuesin between the divergence points [see Fig. 2 (c)]. Whenthe correlation length diverges, the trajectory C crossespoles in the integrand of Eq. (7), thereby preventing thetrajectory from deforming to the real axis. This fact can

4

0.0 0.5 1.0 1.5 2.0γ/4

135ξ

t 4t 4t0.0 0.5 1.0 1.5 2.0

γ/

5101520ξ

0.5 1.0 1.5 2.0

γ/- 400- 200

200400

ξ(a) (b) (c)

FIG. 2. Correlation length ξ [Eq. (10)] for (a) U/4t = 1, (b)U/4t = 0.7, and (c) U/4t = 0.6.

be seen numerically (see red crosses on (off) the trajec-tory C (S) in Fig. 1 (c) [(d)]), and can also be shownanalytically using the Bethe equations [40]. In fact, thesolution with ξ < 0 is not a solution of the Bethe equa-tions, and the analytic continuation from the Hermitiancase breaks down. Similar transitions of Bethe-ansatzsolutions have been found in other non-Hermitian inte-grable models [33, 54, 55].

The poles in the integrand in the first term on theright-hand side of Eq. (7) are given by sin k = λ ± iu.The same condition appears in the construction of thek-λ string excitations in the Hubbard model [21, 56] inwhich a pair of quasimomenta k(1), k(2) form a string con-figuration around a center λ as sin k(1) = λ + iu andsin k(2) = λ − iu. Physically, such string excitations de-scribe the creation of a doublon-holon pair [21]. Theexistence of the poles on trajectory C indicates that thesolution in the thermodynamic limit becomes degeneratewith a k-λ string solution. In fact, the excitation energyof a k-λ string is given by [21, 45]

ε(k) = 2tu+ 2t cos k + 2t

∫ ∞0

dωJ1(ω) cos(ω sin k)e

−uω

ω coshuω,

(11)which vanishes at the poles k = ±π − arcsin(±iu). Herenot only the eigenvalues but also the eigenstates are thesame. This means that the critical point at which thecorrelation length diverges is an exceptional point in thesense that the non-Hermitian Hamiltonian Heff cannotbe diagonalized [24, 57]. Importantly, we can show thatthe nondiagonalizability of Heff leads to the nondiagonal-izability of the Liouvillian L [40]. Thus, the exceptionalpoint is the same for both the non-Hermitian Hamilto-nian and the Liouvillian; however, this does not hold truefor general Liouvillians [58]. Since a nondiagonalizableLiouvillian leads to a singular time dependence of gener-alized eigenmodes [58], the exceptional point significantlyalters the transient dynamics starting from half filling.

The solid curve in Fig. 3 shows the position of the ex-ceptional point as a function of U and γ. Outside theshaded region, the analytic continuation of the Bethe-ansatz solution from the γ = 0 case remains valid. Anincrease of the correlation length in Fig. 2 (b) can be un-derstood as a consequence of the proximity of the systemto the exceptional point. For a large repulsive interac-tion U > 0, a Mott insulator is formed as in the Her-mitian Hubbard model and it has a finite lifetime due

0.0 0.2 0.4 0.6 0.8 1.0 1.2U/4t0.0

0.5

1.0

1.5

2.0

2.5γ/4t

Mott insulatorRe[Δc] > 0

Zeno insulatorRe[Δc] < 0

FIG. 3. “Phase diagram” of the Liouvillian eigenmode thatgoverns the transient dynamics at half filling. The solid curveindicates the location of the exceptional point at which theLiouvillian cannot be diagonalized. The shaded region cannotbe analytically continued from the case with γ = 0. Thedashed curve shows where the real part of the Hubbard gapRe[∆c] vanishes.

to nonzero γ. On the other hand, for small U > 0 andlarge γ, particles are localized due to dissipation. Be-cause the Hubbard gap becomes negative, Re[∆c] < 0,in this region, the localization should be attributed tothe quantum Zeno effect rather than the repulsive inter-action, and therefore this localized state may be calleda Zeno insulator. Interestingly, the phase diagram looksqualitatively similar to that obtained from a mean-fieldtheory for a three-dimensional non-Hermitian attractiveHubbard model [34] after changing the sign of U via theShiba transformation [59].Dissipation-induced spin-charge separation.– Finally,

we address an interesting connection between strong cor-relations and dissipation. The Bethe equations, Eqs. (3)and (4), can be simplified when one takes the large-|u|limit, in which one can expand the equations as (here weset Φ = 0)

kjL = 2πIj +O(1/u), (12)

NΘ

(λαu

)+O(1/u2) = 2πJα +

M∑β=1

Θ

(λα − λβ

2u

).

(13)

These equations indicate that the quasimomenta andspin rapidities are completely decoupled in the |u| → ∞limit [45, 60]. The quasimomenta in this limit are iden-tical to those of free fermions, and Eq. (13) gives thesame Bethe equation as that of the Heisenberg chain af-ter rescaling Λα ≡ λα/u. This leads to a remarkablefact that the Bethe wavefunction is factorized into thecharge part and the spin part [60]. This argument isparallel to that for the spin-charge separation in the one-dimensional Hermitian Hubbard model. However, thecrucial point here is that the spin-charge separation can

5

occur due to large γ even in the absence of the repul-sive interaction U . Thus, in a Zeno insulator, the strongdissipation itself induces a strongly correlated state, andholes created by a loss behave as almost free fermions,whereas the spin excitations are described by a non-Hermitian Heisenberg chain with the exchange coupling4t2/(U − iγ) [35]. As spin-charge separation in a Her-mitian Hubbard chain has recently been observed in ex-periments with ultracold atoms [61, 62], the dissipation-induced spin-charge separation should be observed withcurrent experimental techniques.

Conclusion.– We have shown that the one-dimensionaldissipative Hubbard model is exactly solvable. We haveexploited the integrability of a non-Hermitian Hamilto-nian to diagonalize a Liouvillian using the generic trian-gular structure of Liouvillians of systems with loss [29].We have elucidated how strongly correlated states of theHubbard model are fundamentally altered by dissipation,yet a number of important issues remain open. For ex-ample, the breakdown of the analytic continuation at halffilling suggests that a novel state driven by an interplaybetween strong correlations and dissipation may be real-ized in the shaded region of Fig. 3. Since the standardsolution for the Hermitian Hubbard model cannot be ap-plied to that region, it is worthwhile to investigate thenature of Bethe-ansatz solutions with non-Hermitian in-teractions, as discussed in Refs. [12, 18]. Finally, the so-lution of Liouvillians based on the non-Hermitian Bethe-ansatz method is not limited to the Hubbard model butapplicable to other many-body integrable systems withappropriate Lindblad operators [29]. Examples includeone-dimensional Bose [63, 64] and Fermi [65, 66] gasessubject to particle losses [30], quantum impurity models[67, 68] with dissipation at an impurity [33], and an XXZspin chain [69, 70] with Lindblad operators that lower themagnetization [71]. We expect that the method proposedin this Letter can be exploited to uncover as yet unex-plored exactly solvable models in open quantum many-body systems.

We are very grateful to Hosho Katsura for fruitful dis-cussions. This work was supported by KAKENHI (GrantNos. JP18H01140, JP18H01145, JP19H01838, andJP20K14383) and a Grant-in-Aid for Scientific Researchon Innovative Areas (KAKENHI Grant No. JP15H05855)from the Japan Society for the Promotion of Science.

Note added.– After the submission of this manuscript,a related work [71] appeared in which the Bethe-ansatzapproach to triangular Liouvillians is studied for differentmodels.

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7

Supplemental Material for“Exact Liouvillian Spectrum of a One-Dimensional Dissipative Hubbard Model”

Eigensystem of the Liouvillian

As mentioned in the main text, the eigenvalues of the Liouvillian L are given by λ(N,n)ab = −i(EN,a−E∗N+n,b), whereEN,a and EN+n,b are eigenvalues of the non-Hermitian Hubbard model Heff . The corresponding right eigenoperator

ρ(N,n)ab can be expanded in terms of the basis set {%

(N ′,n)cd = |N ′, c〉R R 〈N ′ + n, d|}N ′,c,d as

ρ(N,n)ab = C

(N,n)ab %

(N,n)ab +

N−2∑N ′=0

∑a′,b′

D(N,N ′,n)aba′b′ %

(N ′,n)a′b′ . (S1)

Note that the basis %(N ′,n′)cd with n

′ 6= n does not appear in the above expansion, since the U(1) charge symmetry ofthe Liouvillian enforces that the Liouvillian does not mix the sectors with different n [41, 42]. Here, we follow Ref. [29]

to determine the coefficients D(N,N ′,n)aba′b′ . We first expand the state in which a right eigenstate |N ′, a〉R is acted on by

the Lindblad operator Lj as

Lj |N ′, a〉R =∑r

v(N ′,a)j,r |N

′ − 2, r〉R , (S2)

where v(N ′,a)j,r = L 〈N ′ − 2, r|Lj |N ′, a〉R under the biorthonormal condition L 〈N ′, a|N ′′, b〉R = δN ′,N ′′δa,b. Then, we

have

J %(N′,n)

ab =∑j

Lj%(N ′,n)ab L

†j

=∑j

∑r,r′

v(N ′,a)j,r (v

(N ′+n,b)j,r′ )

∗ |N ′ − 2, r〉R R 〈N′ + n− 2, r′|

=∑j

∑r,r′

v(N ′,a)j,r (v

(N ′+n,b)j,r′ )

∗%(N ′−2,n)rr′ . (S3)

Substituting Eq. (S1) into the eigenvalue equation Lρ(N,n)ab = λ(N,n)ab ρ

(N,n)ab , we obtain

Lρ(N,n)ab =λ(N,n)ab C

(N,n)ab %

(N,n)ab +

∑j

∑r,r′

C(N,n)ab v

(N,a)j,r (v

(N+n,b)j,r′ )

∗%(N−2,n)rr′

+

N−2∑N ′=0

∑a′,b′

λ(N ′,n)a′b′ D

(N,N ′,n)aba′b′ %

(N ′,n)a′b′ +

N−2∑N ′=0

∑a′,b′

∑j

∑r,r′

D(N,N ′,n)aba′,b′ v

(N ′,a′)j,r (v

(N ′+n,b′)j,r′ )

∗%(N ′−2,n)rr′ , (S4)

λ(N,n)ab ρ

(N,n)ab =λ

(N,n)ab C

(N,n)ab %

(N,n)ab +

N−2∑N ′=0

∑a′,b′

λ(N,n)ab D

(N,N ′,n)aba′b′ %

(N ′,n)a′b′ . (S5)

Since the first terms on the right-hand side of Eqs. (S4) and (S5) are equal, we have∑j

C(N,n)ab v

(N,a)j,a′ (v

(N+n,b)j,b′ )

∗ + λ(N−2,n)a′b′ D

(N,N−2,n)aba′b′ = λ

(N,n)ab D

(N,N−2,n)aba′b′ (S6)

by comparing the coefficient of %(N−2,n)a′b′ . Thus, the coefficient D

(N,N−2,n)aba′b′ is given by

D(N,N−2,n)aba′b′ =

1

λ(N,n)ab − λ

(N−2,n)a′b′

∑j

v(N,a)j,a′ (v

(N+n,b)j,b′ )

∗

C(N,n)ab . (S7)Similarly, by comparing the coefficient of %

(N−4,n)a′b′ , we have

λ(N−4,n)a′b′ D

(N,N−4,n)aba′b′ +

∑j

∑a′′,b′′

D(N,N−2,n)aba′′b′′ v

(N−2,a′′)j,a′ (v

(N+n−2,b′′)j,b′ )

∗ = λ(N,n)ab D

(N,N−4,n)aba′b′ , (S8)

8

and hence

D(N,N−4,n)aba′b′ =

1

λ(N,n)ab − λ

(N−4,n)a′b′

∑j

∑a′′,b′′

D(N,N−2,n)aba′′b′′ v

(N−2,a′′)j,a′ (v

(N+n−2,b′′)j,b′ )

∗

. (S9)Here, we have assumed λ

(N,n)ab − λ

(N ′,n)a′b′ 6= 0 (N ′ = 0, 2, · · · , N − 2), which is necessary for diagonalizability of the

Liouvillian. Repeating the above procedures, all the coefficients D(N,N ′,n)aba′b′ (N

′ = 0, · · · , N − 2) can be obtainedrecursively from C

(N,n)ab .

The left eigenoparator σ(N,n)ab is obtained from the eigenvalue equation L†σ

(N,n)ab = (λ

(N,n)ab )

∗σ(N,n)ab , where the

conjugate of the Liouvillian is defined by [7]

L†ρ ≡ i(H†effρ− ρHeff) +L∑j=1

L†jρLj . (S10)

The left eigenoperator can be expanded in terms of the basis set {ς(N′,n)

cd = |N ′, c〉L L 〈N ′ + n, d|}N ′,c,d by using theleft eigenstates as

σ(N,n)ab = C̃

(N,n)ab ς

(N,n)ab +

Nmax∑N ′=N+2

∑a′,b′

D̃(N,N ′,n)aba′b′ ς

(N ′,n)a′b′ , (S11)

where Nmax = 2L is the maximum value of the particle number of the system. One can determine the coefficients

D̃(N,N ′,n)aba′b′ similarly as for the case of D

(N,N ′,n)aba′b′ shown above. For example,

D̃(N,N+2,n)aba′b′ =

1

(λ(N,n)ab )

∗ − (λ(N+2,n)a′b′ )∗

∑j

(v(N+2,a′)j,a )

∗v(N+2+n,b′)j,b

C̃(N,n)ab , (S12)D̃

(N,N+4,n)aba′b′ =

1

(λ(N,n)ab )

∗ − (λ(N+4,n)a′b′ )∗

∑j

∑a′′,b′′

D̃(N,N+2,n)aba′′b′′ (v

(N+4,a′)j,a′′ )

∗v(N+4+n,b′)j,b′′

. (S13)From Eqs. (S1), (S11), the normalization condition for the eigenoperators reads

Tr[(σ(N,n)ab )

†ρ(N,n)ab ] = (C̃

(N,n)ab )

∗C(N,n)ab = 1. (S14)

The overall coefficients C(N,n)ab and C̃

(N,n)ab in the expansions (S1) and (S11) can be fixed by using this normalization

condition.It follows from the above construction of the eigenoperators ρ

(N,n)ab , σ

(N,n)ab that if the non-Hermitian Hamiltonian

Heff lies at an exceptional point (i.e., it cannot be diagonalized), so does the Liouvillian L. To see this, let us assumethat the non-Hermitian Hamiltonian is parameterized as Heff(g) and that it is at an exceptional point for g = gEP.The eigenvalue equation is given by Heff(g) |N, a, g〉R = EN,a(g) |N, a, g〉R. At the exceptional point, at least twoeigenstates and the corresponding eigenvalues are degenerate:

limg→gEP

(EN,a1(g)− EN,a2(g)) = 0, (S15)

limg→gEP

(|N, a1, g〉R − |N, a2, g〉R) = 0. (S16)

Thus, we have

limg→gEP

(λ

(N,n)a1b

(g)− λ(N,n)a2b (g))

= limg→gEP

(λ

(N−n,n)ba1

(g)− λ(N−n,n)ba2 (g))

= 0, (S17)

limg→gEP

(%

(N,n)a1b

(g)− %(N,n)a2b (g))

= limg→gEP

(%

(N−n,n)ba1

(g)− %(N−n,n)ba2 (g))

= 0, (S18)

limg→gEP

(v

(N,a1)j,r (g)− v

(N,a2)j,r (g)

)= 0, (S19)

9

where λ(N,n)ab (g) = −i(EN,a(g) − E∗N+n,b(g)), %

(N,n)ab (g) = |N, a, g〉R R 〈N + n, b, g|, and v

(N,a)j,r (g) =

L 〈N − 2, r, g|Lj |N, a, g〉R. Then, the above construction of the eigensystem of the Liouvillian shows that the Li-ouvillian eigenoperators corresponding to eigenvalues λ

(N,n)a1b

(g) and λ(N,n)a2b

(g) are degenerate for g = gEP, indicatingthat the Liouvillian lies at an exceptional point. Note that the Liouvllian eigenoperators corresponding to eigenvalues

λ(N−n,n)ba1

(g) and λ(N−n,n)ba2

(g) are also degenerate at g = gEP. Thus, at the exceptional point, the degeneracy of theeigenstates of the non-Hermitian Hamiltonian leads to a large number of degenerate eigenoperators of the Liouvillian.

Proof of the statement on steady states

We show that a steady state of the quantum master equation governed by the Liouvillian L is given by an eigenstateof the non-Hermitian Hamiltonian Heff with a real eigenvalue. Let |Ψ〉 be a right eigenstate of the non-HermitianHamiltonian with a real eigenvalue: Heff |Ψ〉 = E |Ψ〉 , E ∈ R. We assume that the eigenstate is normalized as〈Ψ|Ψ〉 = 1. Then, we have

E = 〈Ψ|H |Ψ〉 − i2

L∑j=1

〈Ψ|L†jLj |Ψ〉 . (S20)

Since H and L†jLj are Hermitian operators, it is required from the real eigenvalue E that

L∑j=1

〈Ψ|L†jLj |Ψ〉 = 0. (S21)

Since 〈Ψ|L†jLj |Ψ〉 ≥ 0, we obtain

〈Ψ|L†jLj |Ψ〉 = 0 (S22)

(j = 1, · · · , L), which is satisfied if and only if Lj |Ψ〉 = 0. Thus, we have

L |Ψ〉 〈Ψ| = 0, (S23)

which shows that |Ψ〉 〈Ψ| is a steady state of the quantum master equation.

Derivation of the dispersion relation of spin-wave excitations

Here, we derive the dispersion relation [Eq. (5) in the main text] of spin-wave-type excitations which provide thelong-lived eigenmodes during relaxation towards steady states. To this end, we consider the Bethe equations (3) and(4) in the main text with M = 1. From Eq. (3), we define the counting function zc(k) as

zc(k) ≡2πIjL

= k +1

LΘ

(sin k − λ1

u

). (S24)

The distribution function ρ(k) is then given by

ρ(k) =1

2π

dzc(k)

dk=

1

2π+

cos k

La1(sin k − λ1)

=ρ0 +1

Lρ̃(k), (S25)

where ρ0 ≡ 1/2π and ρ̃(k) ≡ cos k · a1(sin k − λ1). Here ρ̃(k) gives a 1/L correction to the distribution functiondue to the excitation. The charge quantum numbers {Ij}j specify the distribution of charge excitations. Since thelong-lived eigenmodes are spin excitations, here we assume that the charge quantum numbers take consecutive valuesas Ij = −(N + 1)/2 + j, which express the simplest situation of charge excitations from the Fermi surface. Then, in

10

the large-L limit with a fixed density N/L, the quasimomenta are densely distributed over an interval [−Q,Q], andQ is determined from the particle density as

N

L=

∫ Q−Q

dkρ(k)

=

∫ Q0−Q0

dkρ0 +1

L

∫ Q0−Q0

dkρ̃(k) + 2(Q−Q0)ρ0 +O(1/L2), (S26)

where Q0 = πN/L. Thus, we have

Q−Q0 = −1

L

∫ Q0−Q0

dku cos k

(sin k − λ1)2 + u2+O(1/L2). (S27)

The energy of an excitation is given by

∆E =− 2tL∫ Q−Q

dkρ(k) cos k + 2tL

∫ Q0−Q0

dkρ0 cos k

=− 2t∫ Q0−Q0

dkρ̃(k) cos k − 2t · 2(Q−Q0)L · ρ0 cosQ0

=− 2tπ

∫ Q0−Q0

dku cos k(cos k − cosQ0)

(sin k − λ1)2 + u2, (S28)

and the momentum of the excitation is

∆P =2π

L

N∑j=1

Ij + J1

− 2πL

N∑j=1

(−N

2+ j

)

=−Q0 −1

π

∫ Q0−Q0

dk arctan

(sin k − λ1

u

). (S29)

By eliminating λ1 from Eqs. (S28) and (S29), we obtain the dispersion relation. For ∆P ' 0, the spin rapidity satisfies|λ1| � | sin k|, and hence ∆P ' −Q0 + 2Q0π arctan

λ1u . Thus, for ∆P ' 0, we have

∆E '− 2tπ

∫ Q0−Q0

dk(cos2 k − cosQ0 cos k) ·u

λ21 + u2

'− tπu

(Q0 −

1

2sin 2Q0

)(1− cos π∆P

Q0

), (S30)

which is Eq. (5) in the main text. In Fig. S1, we plot the dispersion relation of the excitations for −Q0 ≤ ∆P ≤ Q0.When one takes the limit of |u| → ∞, the same condition |λ1| � | sin k| holds true. Thus, the dispersion relation(S30) becomes exact for all ∆P in the case of |u| → ∞ (see also Ref. [72]).

Liouvillian spectrum as poles of single-particle Green’s function

The Hubbard gap in the main text is defined from eigenvalues of the non-Hermitian Hubbard model. To clarify itsphysical meaning, we consider a single-particle retarded Green’s function GRj,j′,σ(τ, τ

′) defined by

GRj,j′,σ(τ, τ′) ≡ −iθ(τ − τ ′)〈{cj,σ(τ), c†j′,σ(τ

′)}〉, (S31)

where cj,σ(τ) and c†j′,σ(τ

′) are the Heisenberg representations of fermion annihilation and creation operators, theexpectation value 〈A〉 ≡ Tr[Aρ(0)] is taken over an initial state ρ(0), {A,B} ≡ AB +BA is the anticommutator, andθ(τ) is the Heaviside unit-step function. Suppose that the Liouvillian can be diagonalized. Then, the time evolutionof the density matrix from time τ ′ to τ is expanded in terms of the eigenmodes as

ρ(τ) = eL(τ−τ′)ρ(τ ′)

=∑N

∑n

∑a,b

eλ(N,n)ab (τ−τ

′) Tr[(σ(N,n)ab )

†ρ(τ ′)]

Tr[(σ(N,n)ab )

†ρ(N,n)ab ]

ρ(N,n)ab , (S32)

11

- 1.0 - 0.5 0.5 1.0ΔP

- 0.3-0.2-0.1

Re[ΔE]

- 1.0 - 0.5 0.5 1.0ΔP

- 0.2

- 0.1

Im[ΔE](a) (b)

FIG. S1. Dispersion relation of the spin-wave-type excitations for N/L = 1/3 and u = 0.8−0.5i. (a) Real part of the excitationenergy as a function of the momentum of an excitation. (b) Imaginary part of the excitation energy as a function of themomentum of the excitation. The dots are excitation energies calculated from numerical solutions of the Bethe equations (3)and (4) for L = 240 and N = 80. The black curves give dispersion relations obtained from Eqs. (S28) and (S29). The greencurves show approximate results [Eq. (S30)] that become accurate for ∆P ' 0.

where ρ(N,n)ab (σ

(N,n)ab ) is a right (left) eigenoperator of the Liouvillian with an eigenvalue λ

(N,n)ab . Thus, the single-

particle Green’s function can be decomposed as

GRj,j′,σ(τ, τ′) =− iθ(τ − τ ′)

∑N

∑n

∑a,b

eλ(N,n)ab (τ−τ

′)Tr[cj,σρ

(N,n)ab ]Tr[(σ

(N,n)ab )

†(c†j′,σρ(τ′) + ρ(τ ′)c†j′,σ)]

Tr[(σ(N,n)ab )

†ρ(N,n)ab ]

. (S33)

After the Fourier transformation, we have

GRj,j′,σ(τ′;ω) ≡

∫ ∞−∞

d(τ − τ ′)e(iω−δ)(τ−τ′)GRj,j′,σ(τ, τ

′)

=∑N

∑n

∑a,b

w(N,n)ab (τ

′)

ω − iλ(N,n)ab + iδ, (S34)

which is a counterpart of the Lehmann representation of Green’s function in open quantum systems [50]. Here,

w(N,n)ab (τ

′) ≡Tr[cj,σρ

(N,n)ab ]Tr[(σ

(N,n)ab )

†(c†j′,σρ(τ′) + ρ(τ ′)c†j′,σ)]

Tr[(σ(N,n)ab )

†ρ(N,n)ab ]

, (S35)

and δ = +0 is an infinitesimal positive constant. Substituting the expansions (S1) and (S11) into Eq. (S35), we obtain

w(N,n)ab (τ

′) =1

(C̃(N,n)ab )

∗C(N,n)ab

C(N,n)ab R 〈N + n, b| cj,σ |N, a〉R + N−2∑N ′=0

∑a′,b′

D(N,N ′,n)aba′b′ R 〈N

′ + n, b′| cj,σ |N ′, a′〉R

×

((C̃

(N,n)ab )

∗L 〈N, a| (c†j′,σρ(τ

′) + ρ(τ ′)c†j′,σ) |N + n, b〉L

+

Nmax∑N ′=N+2

∑a′,b′

(D̃(N,N ′,n)aba′b′ )

∗L 〈N ′, a′| (c†j′,σρ(τ

′) + ρ(τ ′)c†j′,σ) |N′ + n, b′〉L

). (S36)

12

Clearly, w(N,n)ab (τ

′) = 0 for n 6= −1. Thus, we arrive at

GRj,j′,σ(τ′;ω) =

∑N

∑a,b

1

ω − EN,a + E∗N−1,b + iδ

×

R 〈N − 1, b| cj,σ |N, a〉R +

N−2∑N ′=0

∑a′,b′

(D(N,N ′,−1)aba′b′ /C

(N,−1)ab )R 〈N

′ − 1, b′| cj,σ |N ′, a′〉R

×

(L 〈N, a| (c†j′,σρ(τ

′) + ρ(τ ′)c†j′,σ) |N − 1, b〉L

+

Nmax∑N ′=N+2

∑a′,b′

(D̃(N,N ′,−1)aba′b′ /C̃

(N,−1)ab )

∗L 〈N ′, a′| (c†j′,σρ(τ

′) + ρ(τ ′)c†j′,σ) |N′ − 1, b′〉L

), (S37)

which shows that the Green’s function has a pole at ω = EN,a − E∗N−1,b. If one takes the limit of γ → 0, the aboveexpression reduces to the standard form for the closed system [50] as

GRj,j′,σ(τ′;ω) =

∑N

∑a,b

〈N − 1, b| cj,σ |N, a〉 〈N, a| (c†j′,σρ(τ ′) + ρ(τ ′)c†j′,σ) |N − 1, b〉

ω − EN,a + EN−1,b + iδ, (S38)

where EN,a and |N, a〉 are an energy eigenvalue and the corresponding eigenstate of the Hermitian Hubbard model.Thus, Eq. (S37) generalizes the standard expression in a closed system, and includes the effect of dissipation in

eigenvalues of the non-Hermitian Hubbard model and in the weight w(N,−1)ab (τ

′). In particular, the real part of theHubbard gap Re[∆c] appears as a gap in the spectral function −(1/π)ImGRj,j′,σ(τ ′;ω) as in the closed system. SimilarLehmann representations can generally be obtained for two-time correlation functions of physical quantities [50]. Thisfact provides a way to experimentally measure the Hubbard gap of the non-Hermitian Hubbard model. For example,as done in cold-atom experiments in Refs. [23, 27, 28], one can prepare a Mott insulating state as an initial state, andlet the dissipative dynamics start to evolve. Since a linear response of the Markovian dynamics to a probe field canbe written with a two-time correlation function [51], the Hubbard gap may be observed in a spectroscopic signal byusing, e.g., lattice modulation spectroscopy [52]. The measurement of the Hubbard gap can also be used to distinguishthe Mott and Zeno insulators as shown in Fig. 3 in the main text.

Dependence of the Hubbard gap on dissipation

In Figs. S2 and S3, we show the dependence of the energy eigenvalue E0 [Eq. (8) in the main text] and that of theHubbard gap ∆c [Eq. (9) in the main text] on dissipation for U/4t = 0.8 and U/4t = 2, respectively. The real partof the energy eigenvalue Re[E0] monotonically increases with increasing the dissipation strength. The absolute valueof the imaginary part of the energy eigenvalue Im[E0] first increases with increasing γ, indicating that the dissipationcauses a decay of the eigenmode. However, |Im[E0]| reaches the maximum at an intermediate dissipation strength, anddecreases for large γ. The decreasing behavior of |Im[E0]| signals the onset of the quantum Zeno effect [26, 27, 46–49].While the qualitative behavior of the energy eigenvalue E0 does not significantly depend on the magnitude of therepulsive interaction U , the Hubbard gap ∆c shows a nontrivial dependence on U . For a weak repulsive interaction[see Fig. S2 (c)], the real part of the Hubbard gap Re[∆c] monotonically decreases with increasing the dissipationstrength, and becomes negative when γ exceeds a certain value. On the other hand, for a strong repulsive interaction[see Fig. S3 (c)], the real part of the Hubbard gap remains positive for any γ. The qualitative difference of Re[∆c] forsmall and large U is attributed to the competition between the repulsive interaction and the quantum Zeno effect.For small U , particles are not well localized in the Mott insulator formed at γ = 0, and therefore the Hubbard gapis significantly affected by localization due to the quantum Zeno effect. On the other hand, for large U , the Mottinsulating state at γ = 0 is not largely changed by dissipation, since particles are already well localized in the Mottinsulator. Therefore, the real part of the Hubbard gap Re[∆c] remains almost unchanged by increasing the dissipationstrength. By contrast, the absolute value of the imaginary part of the Hubbard gap Im[∆c] monotonically increaseswith increasing the dissipation strength [see Fig. S2 (d) and Fig. S3 (d)], because the excitation corresponding to theHubbard gap creates doubly occupied sites and leads to Im[∆c] ∝ −γ for large |u|.

13

0.5 1.0 1.5 2.0 2.5γ/4t

- 0.6- 0.4- 0.2

Re[E0/L](a) (b)

(c) (d)

0.5 1.0 1.5 2.0 2.5γ/4t

- 0.3- 0.2- 0.1

Im[E0/L]

0.5 1.0 1.5 2.0 2.5γ/4t

- 0.5

0.5

Re[Δc]0.5 1.0 1.5 2.0 2.5

γ/4t

- 10- 8- 6- 4- 2

Im[Δc]

FIG. S2. (a) (b) Real [(a)] and imaginary [(b)] parts of the energy eigenvalue E0 as a function of dissipation strength γ forU/4t = 0.8. Dots show numerical solutions of the Bethe equations (3) and (4) for L = N = 2M = 50. The solid curves areobtained from the analytic expression [Eq. (8) in the main text] in the thermodynamic limit. (c) (d) Real [(c)] and imaginary[(d)] parts of the Hubbard gap ∆c as a function of dissipation strength γ for U/4t = 0.8. Dots show numerical solutions of theBethe equations (3) and (4) for L = N = 2M = 50. The solid curves are obtained from the analytic expression [Eq. (9) in themain text] in the thermodynamic limit.

(a) (b)

(c) (d)

1 2 3 4 5γ/4t

- 0.3- 0.2- 0.1

Re[E0/L]1 2 3 4 5

γ/4t

- 0.15- 0.10- 0.05

Im[E0/L]

0 1 2 3 4 5γ/4t

12345Re[Δc]

1 2 3 4 5γ/4t

- 20- 15- 10- 5

Im[Δc]

FIG. S3. (a) (b) Real [(a)] and imaginary [(b)] parts of the energy eigenvalue E0 as a function of dissipation strength γ forU/4t = 2. Dots show numerical solutions of the Bethe equations (3) and (4) for L = N = 2M = 50. The solid curves areobtained from the analytic expression [Eq. (8) in the main text] in the thermodynamic limit. (c) (d) Real [(c)] and imaginary[(d)] parts of the Hubbard gap ∆c as a function of dissipation strength γ for U/4t = 2. Dots show numerical solutions of theBethe equations (3) and (4) for L = N = 2M = 50. The solid curves are obtained from the analytic expression [Eq. (9) in themain text] in the thermodynamic limit.

Divergence of the correlation length at an exceptional point

We follow Ref. [53] to calculate the correlation length ξ as

1

ξ= Im[zc(k∗)], (S39)

14

where

zc(k) = k + 2

∫ ∞0

dωJ0(ω) sin(ω sin k)

ω(1 + e2uω)(S40)

is the counting function derived from the Bethe equations. Here k∗ = π − arcsin(iu) denotes the stationary point ofthe counting function:

dzcdk

(k∗) =1 + 2 cos k∗

∫ ∞0

dωJ0(ω) cos(ω sin k∗)

1 + e2uω

=1−√

1 + u2∫ ∞

0

dωJ0(ω)e−uω

=0. (S41)

Note that k∗ also gives a pole of the integrand in the Bethe equation (7) in the main text. Thus, if the pole k∗ islocated on the trajectory C of quasimomenta, there exists a quasimomentum kj = k∗ that satisfies

zc(k∗) =2πIjL

(S42)

in the thermodynamic limit. Since the quantum number Ij is real, the divergence of the correlation length ξ = ∞occurs as can be seen from Eq. (S39).

Exact Liouvillian Spectrum of a One-Dimensional Dissipative Hubbard ModelAbstract Acknowledgments References Eigensystem of the Liouvillian Proof of the statement on steady states Derivation of the dispersion relation of spin-wave excitations Liouvillian spectrum as poles of single-particle Green's function Dependence of the Hubbard gap on dissipation Divergence of the correlation length at an exceptional point